2009水木艾迪考研数学基础班讲义微积分、线性代数知识点解析（水木艾迪）（2009年）：2009考研数学基础班讲义－微积分第六讲

分类：数学格式：pdf 日期：2009年05月17日

2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
$ ?±s? ? ?s￥à
Q D9

?s￥à
QD?é
? s'à
Q aZED? 1?
M?

à
Q ?sT 1
T￥ Ks? ′? ? ??D′ ? ? ? s
^B?
b

ZEX±s Es? síBE M
9D uWMD b

s?
TD? ?
T￥￡
ü b
?l
?l
!f
μ?>u W
μ?l
?
s ?l
1?w )(xf ],[ ba

?is éuW | ?
:],[ ba
n
xxx,,,
10
L
1?
=?
iii
xxx

i
i
x?= maxλ

?| ],[
1 iii
xx
∈ξ
T
T
∑
=
=
n
i
iin
xfS
1
)(ξ

?K i
O K′D uW sé￥ ?i ?sxfS
n
i
iin
=?=
∑
=
→→
1
00
)(limlim ξ
λλ
],[ ba
[
iii
xx,
1?
∈ ]ξ |′￥?i ?í1
5?f
uW
V
?K′
?1f
u W
￥ s
:T
b
)(xf ],[ ba
sxfS
n
i
iin
=?=
∑
=
→→
1
00
)(limlim ξ
λλ
)(xf ],[ ba
sSdxxfbaI
n
b
a
f
===
→
∫
0
lim)(),(
λ
ba,sY ?1s￥ /a
K
? 1$f

? 1s ?WM

?s￥′
D s?WM
￥?|í1 ' b
)(xf x
∫∫
=
b
a
b
a
dttfdxxf )()(
f
￥ V?Hq
? ? f
μ?>uW V ￥A1 Hq
^f

μ?b ],[ ba )(xf ],[ ba
? ? f
μ?>uW V ￥ s Hq
@/
Hq-B ' V
],[ ba

uW
??μ? )(xf ],[ ba

uW
μ?O oμμK? W?)(xf ],[ ba

uW
??)(xf ],[ ba
> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 1
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
? s?l I ù?￥?¨ ?¨s
Tp+? K
na
è5
? s ￥?é#è¨2

∫∫
=
a
b
b
a
dxxfdxxf )()(

suW￥ V F? $
f
@L??
∫∫∫
+=∈?
b
c
c
a
b
a
dxxfdxxfdxxfRc )()()(,
[]
∫∫∫
+=+
b
a
b
a
b
a
dxxgBdxxfAdxxBgxAf )()()()(

?
V
5)(xf ],[ ba )(xf
9 V
O ],[ ba
∫∫
≤
b
a
b
a
dxxfdxxf )()(

? ?
|?? Vf
],[,0)( baxxf ∈?≥
5 b0)( ≥
∫
b
a
dxxf
? V f
@
5 b)(),( xgxf )()( xgxf ≥
∫∫
≥
b
a
b
a
dxxgdxxf )()(
+Y ?dμ ? ?f

?1
,5 b)(xf ],[ ba 0)( >
∫
b
a
dxxf
w
′? ? ? Vf

@)(xf
],[ ba
Mxfm ≤≤ )(
5
b)()()( abMdxxfabm
b
a
≤≤?
∫
éB ?
?f

dμ V
5
? 11??é )(xg ],[ ba

∫∫∫
≤≤
b
a
b
a
b
a
dxxgMdxxgxfdxxgm )()()()(

s ?′? ? ?f

??

|?|O V

5
)(xf ],[ ba )(xg ],[ ba
),,( ba∈?ξ
P
∫∫
=
b
a
b
a
dxxgfdxxgxf )()()()( ξ
+Y

H
1)( ≡xg ],,[ ba∈?ξ
P
))(()( abfdxxf
b
a
=
∫
ξ
__________
],[
)()(
)(
xff
ab
dxxf
ba
b
a
==
∫
ξ
ü (′
Y
L
VéB ?￡
ü ),,(
0
ba∈?ξ
P

2
? ? b

?

^ V ￥f

5 ?

^)(xf ],[ aa? 0)( =
∫
a
a
dxxf )(xf ],[ aa?
> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 2
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
V ￥
}f

5 b
∫∫
=
aa
a
dxxfdxxf
0
)(2)(

?
^ V￥ ?
ùf

O?
ù 1 T 5?i
L
Aμ
b
)(xf a
∫∫
=
+ TTa
a
dxxfdxxf
0
)()(

? ? ?f
@ 5i )(xf 0)( =
∫
b
a
dxxf ),(
0
bax ∈
P¤ b0)(
0
=xf

￡
üZE ? ?′? ? ￡
üZE ? ??f
￥|?

?d μ ??f
@ 5)(xf 0)( =
∫
b
a
dxxf 0)(],,[ ≡∈? xfbax b

￡
üZE? ??f
￥ |?D s￥|? Q￡
è
! dxxI
∫
=
2
0
1
)sin(sin
π
dxxI
∫
=
2
0
2
)cos(sin
π
5 "

" b
# b
21
1 II <<
21
1 II >>

$ b
%
21
II = 1
21
>> II
}3 ~? )
2
,0(
π
∈x O 1 9f
?
^ xx<sin xsin,sin)sin(sin xx <
dxxI
∫
=
2
0
1
)sin(sin
π
1sin
2
0
=<
∫
dx
π

7 1 hf
5 μ ?
^ xcos xx sin)cos(sin >
dxxI
∫
=
2
0
2
)cos(sin
π
1
2
0
1cos Idxx >=>
∫
π
b
è 9 s ￥ S?
∫
2
0
2
2
dxe
xx
}3 ~
[]
( )
[]
( ) 12min,02max
2
2,0
2
2,0
=?=?
∈∈
xxxx
xx
yN
22
2
0
0
2
0
2
2
0
11
2
=≤≤=
∫∫∫

dxedxedxee
xx

è
! dxxxxM )1(ln
22
1
1
++=
∫
dx
x
xx
N
∫
+
+
=
1
12
3
1

∫
+
=
1
1
22
3
)1(
1
dx
x
x
P
5 "
b
"
b #
b $
NMP << PNM << NPM << b %
MPN << b
}3 ~?? M 1f
? uW￥s #1
> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 3
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
0)12(212
1
2
1
0
1
0
2
2
2
>?=+=
+
=
∫
x
x
dx
N 0
)1(
1
2
22
1
0
<
+
=
∫
dx
x
P
[ bNMP <<

di ??
D%
TD?s￥9

d i ??
D%
T#?¨
? ?
d i ??
D%
T
?
^
￥ ? ?f

1
￥B?ef

5 i è
C

P

)(xf ],[ ba )(xF )(xf ],[ ba
CdttfxF
x
a
+=
∫
)()( ],[ bax∈?
b
a
b
a
xFaFbFCbFdxxf )()()()()(
=?=?=
∫

T?1
d i ??
D%
T+ Y
μ
b)()()( afbfdxxf
b
a
=′
∫
d i ??
D%
T
P¤? s￥9
1p? ?sù5 pef
ù5b
?¨
di ??
D%
Tá
ì V[ YV?? sp￥? s￥′b B? V°¤¨
X± sEaDí Es? sE9
? sb
è p
∫
b?
2
0
|1| dxx
}3 ~
∫∫∫
+?=?
2
1
1
0
2
0
)1()1(|1| dxxdxxdxx
1
22
2
1
2
1
0
2
=
+
= x
xx
x
? ?s
? l￥f
?s9
?+Y?i s
sb
è p,sin1
0
∫
π
dxx
}3 ~
∫∫
=?
ππ
00
2
cos
2
sin.sin1 dx
xx
dxx

∫∫
+
=
π
π
π
2
2
0
2
cos
2
sin
2
sin
2
cos dx
xx
dx
xx
.424?=
è
!
p b
>+
≤?
=
01
01
)(
xx
xx
xf
∫
1
1
)( dxxf
}3~3 EB uW
=μ ?B ? W ?
y N)(xf ]1,1[? ]1,1[? u W
=? ie
f

?
°¤ ¨
di ? ?
D%
T ?¨u W￥ VF? μ
> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 4
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
)( dxx
0[],0,1[?
)(
0
1
=
∫
dxxf
2
3
1
0
=

#
∫
1
1
(f
3E = (f
è p
}3 ~
∫
2
2
1
1(
π
π
M
9
?BDíE￥'
±
^
X±sZE

b
a
xf ))((?′
∫
?=DíE￥
)( dxxf
b
a
=
∫
μQf
t =
Dí E￥×1?
7
f
b
a
∫
7
μ
<W
> ?

6;
∫∫∫

+=
1
0
0
1
1
1
)()( fdxxfdxxf
= sY V[¨
di ??
D%
T
]1,
2
3
2
0
1
2
=
x
x

2
)(
2
1
0
+=
∫
x
x
dxxf
b)dxx
^)x ]1,1[? ￥f

∫
1
1
)( dxxf
xx+
2
cos)1(
π
dx
x
∫
+
2
2
cos1
π
b
dx
x
x
dx
x
xx
∫
+
=
+
+
2
0
22
cos1
cos
2
cos
cos)
π
12
12
ln
2
1
sin2
)(sin
2
2
0
2
+
=
=
∫
dx
x
xd
π

DE
)()()( afbfdxxf
b
a
=′
∫
b
a
xfdxx ))(()( =′?
'
±
^
β
α
β
α
))(()())(( tFdtttf =′?
∫
? 1 p D)(xf )(t?′ ?? )(tx?=

O)(
1
x
)(),( β?α? == ba CxFdxxf +=
∫
)()( b
¨-B
^u WMD[ ?Msu W1+?
" ￥￥MD

∫
=
b
a
f
dxxfbaI )(),(
dttxtxfdxxf
b
a
)())(()(
1
0
′?
∫∫

ab
ax
t
=
dttxtxfdxx
d
c
)())(()( ′?
∫

ccd
ab
ax
t +?
= )(
Q|MD xt?= ?
MD
x
t
1
= b
¨?s￥iD·s b

I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 5
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055

è p
∫
3
4 2
4x
dx
b
}3 ~
7
xu?=
∫
3
4 2
4x
dx

∫
4
3 2
4u
du

7
5
tu sec2= tdtdu sectan2=
?
H3=u
3
2
arccos=t
?
H4=u
3
π
=t

=
∫
3
4 2
4x
dx
∫∫
=
3
3
2
arccos
4
3 2
tan2
sectan2
4
π
dt
t
tt
u
du

∫
=
3
3
2
arccos
2
cos
cos
π
dt
t
t
∫
+
+
=
3
3
2
arccos
sin
sin1
1
sin1
1
2
1
π
td
tt

7
V°¤ ? ¨X±sE
ty sin=
∫
+
+
3
3
2
arccos
sin
sin1
1
sin1
1
2
1
π
td
tt

2
3
3
5
2
3
3
5
1
1
ln
2
1
1
1
1
1
2
1
u
u
dy
yy?
+
=
+
+
=
∫

2ln)53ln()32ln( ++?+= b
è
! () ( )
∫
=?
4
0
2
2cos
π
dxxfxxf p ()
∫
2
0
π
dxxf b
}3 ~,() Idxxf =
∫
2
0
π

7 ux =2 5 dudx
2
1
=
()=
∫
4
0
2
π
dxxf () Iduuf
2
1
2
1
2
0
=
∫
π
b
?
T () ( )
∫
=?
4
0
2
2cos
π
dxxfxxf
H| s¤?

() Idx
I
dxxxf
42
]cos[
2
0
2
0
2
π
ππ
==?
∫∫

' IdxxI
4
cos
2
0
2
π
π
=?
∫

> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 6
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
# dxxII
∫
=?
2
0
2
cos
4
π
π
422
1 ππ
=?=

yN ==
∫
dxxfI
2
0
)(
π
π
π
4
b
è
!
^
￥ ??f

5
%b)(xf ],[10
"

∫∫
=
ππ
π
00
)(sin)(sin dxxfdxxxf
#

∫∫
=
ππ
π
00
)(sin2)(sin dxxfdxxxf
$

∫∫
=
ππ
π
00
)(sin
2
)(sin dxxfdxxxf
%

∫∫
=
ππ
π
00
)(sin
2
)(sin dxxfdxxxf
}3 ~
7 dtdxtx?=?=,π

∫∫
=
0
0
)(sin)()(sin
π
π
π dttftdxxxf

∫∫
+?=
ππ
π
00
)(sin)(sin dttfdtttf
M[ ¤?s?1 %b

s? sE
! D ?? 1
￥B? ef
5 )(xf )(xg′ ],[ ba )(xF )(xf ],[ ba
∫∫
′?=
b
a
b
a
b
a
dxxgxFxgxFdxxgxf )()()()()()(
è p
∫
e
e
xdx
1
ln
}3 ~ ()
∫∫
=
e
e
e
e
e
e
xxdxxxdx
1
1
1
lnlnln
e
dx
e
e
e
e
21
1
=?
=
∫
b
è ￡
ü
∫
2
0
sin
π
xdx
n
∫
=
2
0
cos
π
xdx
n
ip
∫
=
2
0
sin
π
xdxJ
n
n
b
}￡ ~
7 tx?=
2
π

∫
=
0
cos
π
tdtJ
n
n
2
∫
=
2
0
cos
π
xdx
n
b
∫∫
==
22
0
1
0
)cos(sinsin
ππ
xxdxdxI
nn
n

> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 7
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
))(1(
cossin)1(cossin
2
0
22
2
0
1 2
nn
nn
IIn
xdxxnxx
=
+?=

∫
ππ

2
1
=
nn
I
n
n
I
′L,3,2=n 1,
2
10
== II
π
b
?

2T? 1s￥? w
T è¨?w
T μB??w =??w ì
T ?·
￥
^ [?w
TV
Us 2T A? ó′ B??wì
T3μB? ′ =? ?
wì
T3μ=? ′?
??!￥ 9
ì
Tb

2T VB
,¤?/
L ¨?
T
1
!)!12(
!)!2(
,
2!)!2(
!)!12(
122
+
=?
=
+
n
n
I
n
n
I
nn
π

L,3,2,1=n b
è =
+?
=
∫
dx
xee
I
xx
2
0
5cossin
8
sin
π

b
"

4
π
b #

15
1
b $

8
π
b %

30
1
b
}3 ~? ?? Dsà
Q ?'¤? s?
15
1
3
2
5
4
8
1
==I ê #

è X? dt
t
e
A
t
∫
+
=
1
0
1
5 =
+
∫
dt
t
e
t
1
0
2
)1(
s? A
e
+?
2
1 b
}3 ~ y1 dt
t
e
A
t
∫
+
=
1
0
1

[
∫∫
+
+
+
=
+
1
0
1
0
1
0
2
11)1(
dt
t
e
t
e
dt
t
e
ttt

A
e
+?=
2
1 b
è
! 1? ?

9
n
∫
1
0
2
)1( dxxx
n
}3 ~
ZE
∫
1
0
2
)1( dxxx
n
dxxx
nn
xx
n
n
∫
+
+
+
+
+
=
1
0
1
1
0
12
)1(
1
2
1
)1(

dxx
nnnn
xx
n
n
∫
+
+
++
+
++
=
1
0
2
1
0
2
)1(
)2)(1(
2
)2)(1(
)1(
> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 8
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
)3)(2)(1(
2
)3)(2)(1(
)1(2
1
0
3
+++
=
+++
=
+
nnnnnn
x
n

ZE
7 dtdxtx?==?,1 5 μ
∫
1
0
2
)1( dxxx
n

3
1
2
2
1
1
)1(
1
0
2
+
+
+
+
=?=
∫
nnn
dttt
n

MKs
M
K s
!
V
i· B￥
L
D- ?
y N M

K s ?l
B ?f

:T á
ì? 1M
K sb
)(xf ],[ ba ],,[ bax∈?
∫
x
a
dttf )(
∫
x
a
dttf )(
∫
=
x
a
dttfxF )()(
? ?
?
V
5M
K s ?l￥f

? ?b
)(xf ],[ ba
∫
=
x
a
dttfxF )()(
],[ ba
? ? B?
^
￥ef
b)(xF )(xf ],[ ba

?
??
5M
K s ?l￥f

V
?
O
)(xf ],[ ba
∫
=
x
a
dttfxF )()( ],[ ba
)()( xfdttf
dx
d
x
a
=?
∫
b
?i B?
^
￥B?ef
b)(xF )(xf ],[ ba
}￡~
],,[ bax∈? 5

∫
=
x
a
dttfxF )()(
∫∫∫
+?+
=?=?
xx
x
x
a
xx
a
dttfdttfdttfxF )()()()(
y1
V
y N
μ ?')(xf ],[ ba )(xf ],[ ba ],,[ bax∈? i
P¤0>M
Mxf ≤)( ?
^
xMdttfxF
xx
x
≤≤?≤
∫
+
)()(0
?C /? ? V¤ yN
??0)(lim
0
=?
→?
xF
x
∫
=
x
a
dttfxF )()( ],[ ba

?
∫
x
a
dttf
dx
d
)(
=
∫
+
→?
xx
xx
dttf
x
)(
1
lim
0

→? aax
x
0
=
∫∫
+ xxx
dttfdttf )()(
1
lim
∫
+
→?
=
xx
xx
dt
x
f )(
lim
0
ξ
> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 9
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
)()(lim
)(
0
xff
x
x
==
→
→?
ξ
ξ

T ? ξ1 Dx xx?+ -W ￥B?

￡
ü¨ ?s?′ ? ? b
è
! ? dttfxF
x
∫
=
0
)()(
≤<?
≤≤?
=
.20,2
,01,
)(
2
2
xx
xe
xf
x
5
=<%> b)(xF ]2,1[?
"
^ ￥ ef
#
V? )(xf
$
? ?? %
??? ? V?b
}3 ~
^)(xf ]2,1[?
μ ?B ?W ?￥ Vf

à μef
b
? M/Ks μ ]"￥? ébO
∫
b
x
dttf )( )()( xfdttf
dx
d
b
x
=?
∫

ˉMKs
∫
)(
)(
)(
x
x
dttf
β
α
? ?
?
V
)(xf ],[ ba )(),( xx βα
??5ˉMKs
^ ? ?f
b
],[ ba
∫
)(
)(
)(
x
x
dttf
β
α

?
??)(xf ],[ ba )(),( xx βα
V?
5ˉM Ks
^ V ?f

?
1
],[ ba
∫
)(
)(
)(
x
x
dttf
β
α
)())(()())(()(
)(
)(
xxfxxfdttf
dx
d
x
x
ααββ
β
α
′?′=
∫

MK s￥M1ù5
;? MKs? l
B?f

*
1 B í ±s?f
￥
μ
MK s9]"
V[ é?b è? á
ì V[ ) ?/
ù 5
M K s?l￥f
T1í k l
¨￥ 9
A r E5M K s?l ￥f
K ù5?￥? ¨b
MKs? l￥f
￥ ???# ′ù5
MK s?l￥f
￥
ü à Z 7
MK s?l￥f
￥s ù5
1? ef
￥B t×12

2
??f
-ef
A1
}f
b
2
??
}f
-ef
A1f
Dè
- ?oμB ?1f

b0=C
2
???
ùf
-ef
A1?
ùf
DL?
-O ?
ù?Mb
?? ?
ùf
- ef
1?
ùf
￥ 1Hq
^ )(xf
> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 10
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
0)(
0
=
∫
T
dxxf ? 1?
ùb 0>T
2
μ?B ? W?￥f
àμef
b
2
μ?= ? W?￥f
V[μef
b
2
MKs V
U￥f
?B?
^ef
b

è￡
ü ??
}f
- ef
A1f
Dè
-b
}￡ ~:
∫
=
x
a
dttfxF )()(
1￡ if
P¤)(xG CxGxF += )()(
? 1è
b )()( xfxF =′ C
! 1￡
@
∫
=
x
dttfxG
0
)()( )(xG
)()( xGxG?=?
A? ?CxGxF += )()( )()( xfxf =? y N
)( xG?
∫
=
x
dttf
0
)(
7 ?
^¤? tu?=
∫
=?
x
udufxG
0
)()()(
∫
=
x
duuf
0
)( )(xG?= b
è
!
^ ? ?￥?
ù1 T ￥f

￡
ü )(xf
∫∫
=
+ TaT
a
dxxfdxxf
0
)()(
}￡ ~￡E By1
^ ??f

| aT 1M

)(xf
0)()()( =?+=
∫
+
afaTfdxxf
da
d
aT
a

∫
+aT
a
dxxf )(1? M
1è

7?a 0=a
H

Cdxxfdxxf
TaT
a
==
∫∫
+
0
)()(

yN ?i￥ ( ? ?Cdxxfdxxf
TaT
a
==
∫∫
+
0
)()( a
￡E =

∫
+aT
a
dxxf )(
> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 11
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
∫∫∫
+
++=
aT
T
T
a
dxxfdxxfdxxf )()()(
0
0

∫
∫∫∫
=
++=
T
aT
a
dxxf
dxTxfdxxfdxxf
0
00
0
)(
)()()(
? ￡EB1p
^ ??f

￡E=? 1p
^ Vf
)(xf )(xf
è
! dttxf
x
2
cos1
0
sin)(
∫
=
65
)(
65
xx
xg += 5 ?
H
^
￥
b
0→x )(xf )(xg
"
?¨íkl
b #
ú ¨íkl
b $
?Níkl
b %
] ¨?d?Níkl

s? #

è
! ),0[)(),( +∞∈Cxgxf ??9F50)( >xf )(xg
dttf
dttgtf
x
x
x
)(
)()(
)(
0
0
∫
∫
=? <>
"

??9F b #
),0[ +∞ ),0[ +∞
??h

$

??9 F
? ?h
b )1,0[ + ),1[ +∞
%

??h

? ?9Fb )1,0[ + ),1[ +∞
}3 ~??
2
0
00
)(
)()()()()()(
)(
=′
∫
∫∫
dttf
dttgtfxfdttfxgxf
x
x
xx

,0
)(
)]()()[()(
2
0
0
>
=
∫
∫
dttf
dttgxgtfxf
x
x

[ )(x?
??9F
s ? "
),0[ +∞
è
! [
??
￡
üf
,0)( >xf ]
]
ba,[]
∫∫
+=
x
b
x
a
dttfdttfxF
1
)()()(
[
μO?μB ?
,?b ba,
> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 12
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
}￡~
??
5 ,0)( >xf [ba,] )(xF [ ]ba,
??O V?
0
)(
1
)()( >+=′
xf
xfxF
)(xF [
??9?y1 ]ba,
0)()(,0
)(
1
)( >=<=
∫∫
b
a
a
b
dttfbFdt
tf
aF ? ??f
,?? ? V?

μ O?μB?
,?b
)(xF
[ba,]
è ? ?è
￥′
P cba,,
0
)1ln(
sin
lim
3
0
≠=
+
∫
→
c
dt
t
t
xax
x
b
x

}3 ~
n5?s
K1
,Aμ
0
)1ln(
lim
3
0
=
+
∫
→
x
bx
dt
t
t
b?
^ b0=b
Q
∫
+
→ x
b
x
dt
t
t
xax
)1ln(
sin
lim
3
0

)1ln(
)cos(
lim
)1ln(
cos
lim
3
0
3
0
x
xax
x
x
xa
xx
+
=
+
=
→→
0
cos
lim
2
0
≠=
=
→
c
x
xa
x

Aμ
2
1
,1 =?= ca b
è
!
V?)(xf ),0[ ∞+ 0)0( =f Qf
1
p b
)(xg
x
xf
exdttg
2
)(
0
)(
∫
=
)(xf
}3 ~??
T
§1?
∫
=
)(
0
2
)(
xf
x
exduug
xp?
?i? ¤? xxfg =))((
,2)(
2 xx
exxexfx +=′ K? 0≠x 5μ
xx
xeexf +=′ 2)(
s ¤? ?C)1()( ++=
x
exxf 0)0( =f #
)0(1)(lim
0
fCxf
x
=+=
+
→
3 1?=C ?
^ b1-)1()(
x
exxf +=
è
! 1 ? ?dμf

μv ? ￥è
C?)(xf bx≤≤1 # 0 )(xfy ≤≤
> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 13
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
?? u×￥
1 21
2
+b p b)(xf
}3 ~? 21)(
2
1
+=
∫
bdxxf
b
¤ 21)(
2
1
+=
∫
xdttf
x

|

T
H p ?¤x
1
)(
2
+
=
x
x
xf b
è X ?
1
}f
O 5
^<" >
)(xf ),( +∞?∞ dttfttxF
x
∫
=
0
)()2(sin)(
)(xF
"
}f
#
f
$
?
ùf
%
[
??f
??
^
cμ?
￥ s
s|
=cμ?
￥ù5
^ B ?×15 ?? ?ù 5aa31 ?
p?
b
?? ZEμ
?

??
[y 0?
TC s|
=
H5|c ?
￥y0 M?s| ?

?í E|c ?
￥?s M?s| ?
5? ?M
9D
uWMD b
è =?
∫
x
dttx
dx
d
0
2
)2cos( b
}3 ~
7 5 dudtutx?==?,2
∫∫
=?=
x
x
x
x
duuduuxI
2
2
2
2
coscos)( b =′ )(xI
22
cos)2cos(2 xx? b
è X? ??f
@?
T p ￥V r
Tb )(xf
2
1
0
)(2)( xxfdttxf?=
∫
)(xf
}3~ y 1
n5) ? ?

7
x
du
dtutx ==,?
^
∫∫
=
x
du
x
ufdttxf
0
1
0
1
)()(

3
0
)(2)( xxxfduuf
x
=
∫
H1? p?i ? ?¤ b x
2
3)()(2 xxfxfx =+′
?
H 30≠x xxf
x
xf
2
3
)(
2
1
)( =+′ ¤ )
5
3
(
1
)(
2
xxC
x
xf +=
?? O
[0)0( =f )0()(lim
0
fxf
x
=
→
0=C '
2
5
3
)( xxf = b
è
! 1 X? V? f
1 ￥Qf
5)(xf )(xg )(xf
=?
∫
)(
)(
xfx
x
dtxtxg
dx
d
"

"
# )()(
2
)(
0
xfxdttg
xf
′+
∫
)()(
2
)(
0
xfxdttg
xf
′?
∫
> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 14
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055

$
% )()(
)(
0
xfxdttg
xf
′+
∫
)()(
)(
0
xfxdttg
xf
′?
∫
}3 ~:
∫∫

=?=
)()(
)()()(
xfx
x
xfx
x
dtxtgxdtxtxgxI

T?￥s
7
dudtuxt ==?,5
∫
=
)(
0
)()(
xf
duugxxI ))())((()()(
)(
0
xfxfxgduugxI
xf
′+=′
∫

))())((()(
)(
0
xfxfxgduug
xf
′+=
∫
"
)()(
2
)(
0
xfxduug
xf
′+=
∫
è =
+
∫
+∞→
x
x
t
x
dt
e
t
2
1
ln
lim b
}3 ~s ? 0b
P ?
H? ?f

? é
P?
H μ
0>?X 0>> Xx xe
x
ln,0>?X
0>> Xx
x
x
x
x
x
x
t
e
xx
dt
e
x
dt
e
t
+
=
+
<
+
<
∫∫
1
2ln
1
2ln
1
ln
0
22

?¨ C/? ?¤ ? 0
1
ln
lim
2
=
+
∫
+∞→
x
x
t
x
dt
e
t
b
s 8 è5
è X ? ? ? w L )(xfy =1? ? )0()0,( ≠aa ?
5

%b
∫
∈?
c
c
dxxafRc )(,

" #
$
%
dxxaf
c
∫
0
)2(2 dxxaf
c
∫
0
)2(2 dxxcf
a
∫
0
)(2 0
}3~?+il V¤?ê %
w L )(xfy =1? ? )0()0,( ≠aa ?b
[/
^+?¨? uWMD￥ è5b
è X ?
￥ ??wL],[ ba )(xfy =1?°L
2
ba
x
+
= ?
￡
ü
∫∫
+
=
2
)(2)(
ba
a
b
a
dxxfdxxf
}￡ ~
∫∫∫
+
+
+=
b
a
b
a
ba
ba
dxxfdxxfdxxf
2
2
)()()(
??1 ?°L)(xfy =
2
ba
x
+
= ?aB sμ
∫
+
b
ba
dxxf
2
)(
∫
+
+=
b
ba
dxxbaf
2
)(
7 5 ?
^¤ ?xbat?+= dtdx?=
∫∫
++
=?+
a
ba
b
ba
dttfdxxbaf
22
)()(
∫
+
=
2
ba
a
dttf )(
[
∫∫
+
=
2
)(2)(
ba
a
b
a
dxxfdxxf b
> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 15
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
è p ￥K vKl′b dtetxf
x
t
∫
=
2
0
)2()(
}3~ 1
}f
o3p)(xf ),[ +∞0
￥Kv Kl′b
7
2,0)2(2)(
2
2
==?=′
xexxxf
x
1· B?O
? 20 << x
H ,0)( >′ xf
? 2>x
H yN,0)( <′ xf 2=x 1 v′?' Kv′?b
Kv ′1
2
2
0
1)2()2(

+=?=
∫
edtetf
t
b
=
+∞→
)(lim xf
x
dtet
t
∫
∞+
0
)2( 112)2(
00
=?=+=
+∞
+∞
tt
eet
? y N 1Kl ′?Kl ′1 b0)0( =f 0=x
è
!
V ±)(xf ],[ ba )(xf
′
dh ￡
ü )]()([
2
)( bfaf
ab
dxxf
b
a
+
≤
∫
b
}￡ ~M[ /￡ ùf

!],[ bax∈?
∫
+
=
x
a
dttfxfaf
ax
xF )()]()([
2
)(

0)( =aF
5o?ù ? ￥? ??
￡
ü)(xF 0)()( =≥ aFxF b
)()(
2
)]()([
2
1
)( xfxf
ax
xfafxF?′
++=′ )(
2
)]()([
2
1
xf
ax
xfaf ′
+?=
)(
2
)(
2
)(
xf
ax
f
xa
′
+′
= ξ
)]()([
2
ξfxf
ax
′?′
=
),( xa∈ξ b
)(xf ′ dh
?
0)( ≥′ xF 0)()( =≥ aFxF b
è ￡
ü
∫∫
+
≤
+
22
00
22
1
cos
1
sin
ππ
dx
x
x
dx
x
x

}￡ ~
Z E
∫∫
+
+
+
=
2
4
4
2
0
2
1
cossin
1
cossin
π
π
π
dx
x
xx
dx
x
xx
I

?=? s
7 xt?=
2
π
5μ
∫
+
=
2
0
2
1
cossin
π
dx
x
xx
I
∫∫
+
+
+
=
4
0
2
4
0
2
)
2
(1
sincos
1
cossin
ππ
π
dt
t
tt
dx
x
xx
∫
+
+
=
4
0
2
2
]
)
2
(1
1
1
1
)[cos(sin
π
π
dt
t
t
tt
> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 16
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
0
)
2
(1)1(
)
4
)(cos(sin
4
0
22
2
<
++

=
∫
π
π
π
π
dt
tt
ttt

ZE
∫
+
=
2
0
2
1
cossin
π
dx
x
xx
I
∫∫
+
+
+
=
2
4
2
4
0
2
1
cossin
1
cossin
π
π
π
dx
x
xx
dx
x
xx
I
∫
+
=
4
0
2
1
)cos(sin
1
1
π
ξ
dxxx
∫
+
+
2
4
2
2
)cos(sin
1
1
π
π
ξ
dxxx
? )
2
,
4
(),
4
,0(
21
ππ
ξ
π
ξ ∈∈

? =?s
7 xt?=
2
π
5μ
∫
+
+
=
4
0
2
2
2
1
)cos(sin)
1
1
1
1
(
π
ξξ
dxxxI
0)21)(
1
1
1
1
(
2
2
2
1
<?
+
+
=
ξξ

è Z ? 01
0
2
cos0
2
2
=++
∫∫
x
t
x
dtedtt
π
￥?￥?
1 #
b
"
#
b $
b %
b
}3~
! dtedttxf
t
x
x 20
2
cos
2
0
1)(
∫∫
++=
π
μ 0)0(
0
1
2
<=
∫
dtef
t
01)1(
1
0
2
>+=
∫
dttf
[ )1,0(∈?ξ
P 0)( =ξf ?
0
2
sin
2
1)(
2
cos
2
2
>?++=′
x
e
x
xxf
π
ππ

[μO?μ B??b
è
!
??
= V?O
@1>k )(xf ]1,0[ )1,0(
0)()1(
1
0
1
==
∫
dxxfxekf
k x

￡
ü à
iB ? )1,0(∈ξ
P¤ b)()1()(
1
ξξξ ff
=′
}￡~ ?s?′ ? ?¤ )()1(
1
1
1
1
ξξ
ξ
fef
= )
1
,0(
1
k
∈ξ
! )()(
1
xfxex
x?
=?
> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 17
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
5 )(x? ]1,[
1
ξ
?? )1,(
1
ξ
= V?
O )1()1()(
1
f==?ξ? yNi )1,0()1,(
1
∈ ξξ
P¤
0)()()()(
111
=′+?=′

ξξξξξξ?
ξξξ
fefefe ?
^0
1
≠
ξ
e
0)()()( =′+? ξξξξξ fff
[ b)()1()(
1
ξξξ ff
=′
è
! ()
∫
+
=
2
,sin
π
x
x
dttxf
￡
ü

^[()xf π1 ?
ù￥?
ùf

p ( )xf ￥′ ×b
}3~
()
∫
+
+
=+
π
π
π
2
3
,sin
x
x
dttxf
! π+=ut 5μ
()
∫
+
+=+
2
)sin(
π
ππ
x
x
duuxf
∫
+
==
2
)(sin
π
x
x
xfduu
#
^[()xf π1?
ù￥ ?
ùf
b

y 1 xsin ( )+∞∞?,
? ? ?i? ( )xf ￥ ?
ù1 π#o3 []π,0
)
′ ×by1 () )sin()cos()sin()
2
sin( xxxxxf?=?+=′
π

7 ¤() 0=′ xf,
4
3
,
4
21
ππ
== xx O
∫
==
4
3
4
2sin
4
π
π
π
tdtf
∫∫∫
=?==
π
π
π
π
π
π
π
4
3
4
5
4
5
4
3
,22sinsinsin
4
3
tdttdtdttf ?
() () ( )
∫∫
=?===
2
0
2
3
,1sin,1sin0
ππ
π
π dttftdtf y7 ( )xf ￥ Kl′
^ 22? Kv′
^ 2 # ￥′×
^()xf [ ].2,22? b
> ?

6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 18

课件简介

课件名称：	2009水木艾迪考研数学基础班讲义微积分、线性代数知识点解析（水木艾迪）（2009年）
课件分类：	数学
课件类型：	教学课件
文件大小：	2.83MB
下载次数：	30
评论次数：	12
用户评分：	6.8